Gödel and the Limits of Mathematics
In 1931, an Austrian logician called Kurt Gödel published the ‘Incompleteness Theorems’ which proved two fundamental limitations in maths. First, they proved that there are mathematical truths which can’t be formally proven and second, that maths itself can’t be proven to be internally consistent using its own fundamental assumptions and deductive rules.
For about fifty years prior to the publication of the Incompleteness Theorems, mathematicians had been trying to establish secure foundations upon which to build mathematics. Gödel showed that this isn’t possible and, in doing so, undermined the faith in the certainty of mathematical truths that mathematicians had held for over 2,000 years.
A decade after publishing the Incompleteness Theorems, Gödel fled Nazi-occupied Austria and emigrated to America where he worked at the Institute for Advanced Study (IAS) in Princeton. At the IAS, he became close friends with Albert Einstein and they went for long walks together discussing maths, science and philosophy. Einstein later said the main reason he went into work at the IAS was to be able to walk home with Gödel. A few years before moving to America, Gödel became paranoid about being poisoned and refused to eat any food that hadn’t been prepared by his wife.1 In 1977, she was taken into hospital and whilst she was away and unable to make his food he stopped eating and eventually starved to death.
The Essence of the Incompleteness Theorems
The Gödel Sentence
Gödel proved both Incompleteness Theorems using the ‘Gödel Sentence’, G, which states: ‘G can’t be logically proven to be true.’
The First Incompleteness Theorem
Consider the Gödel Sentence, G, and its negation, ~G:
A logical proof that G is true would prove that ~G is true. If a logical system is consistent, it can’t be used to prove that both a statement and its negation are true. Therefore, in a consistent logical system, G can’t be proven. However, that’s exactly what G claims. In other words, the statement G, which states ‘G can’t be logically proven to be true’, can’t be logically proven to be true. So, G is substantially true but its truth can’t be proven in a consistent logical system. Therefore, there are true logical statements which can’t be proven logically in a consistent logical system.
Gödel’s Second Incompleteness Theorem
Consider G and the First Incompleteness Theorem:
The First Incompleteness Theorem states ‘In a consistent logical system, G can’t be logically proven to be true.’ However, the statement ‘G can’t be logically proven to be true’ is the statement G itself. So, the statement ‘In a consistent logical system, G can’t be logically proven to be true’ is equivalent to the statement ‘In a consistent logical system, G is true’.
Suppose there is a proof that the system is consistent. That would give us proofs of the statements ‘In a consistent logical system, G is true’ and ‘This logical system is consistent’. These two statements prove the statement ‘G is true’. However, as we saw above, if G is true then ~G is also true, which proves that the system isn’t consistent. Therefore, a logical system can’t be proven to be consistent using its own logical rules.
The Significance of the Incompleteness Theorems
The Incompleteness Theorems proved that it’s impossible to create a secure logical foundation for maths, which had been developing steadily for over 2,000 years.
A Short History of Logic and Maths
Logic is a system for deriving theorems from a set of axioms and inference rules. Axioms are propositions that are assumed to be true and inference rules are the types of logical deduction which are valid. A statement which can be derived from the axioms using the inference rules is a theorem. This system was discovered by the ancient Greeks and the mathematician Euclid used it codify the axioms and inference rules of geometry. The ability of only a few axioms and inference rules to support a whole branch of maths containing millions of theorems led most mathematicians over the next 2,000 years to believe that similar foundations could be found for all of maths.
However, one of the geometric axioms that Euclid proposed was ‘Given a line and a point outside that line, there is only one line which runs through the point and parallel to the line’. Euclid chose his axioms because they seemed self-evident but the parallel axiom seemed less obvious that the others to the ancient Greeks, so they tried unsuccessfully to derive it from the other axioms. Over 2,000 years later, it was finally proven that the parallel axiom can’t be derived from the rest of Euclid’s axioms, so mathematicians experimented by replacing it with different axioms which stated that either more than one parallel line can be drawn or that none can be drawn, and then used those axioms to prove theorems in the resulting ‘non-Euclidean’ geometric systems.
It was generally believed that because the axioms of Euclidean geometry are true of the physical world, the theorems of Euclidean geometry must be objectively true. However, the inability to derive the parallel axiom and the subsequent development of non-Euclidean geometry undermined this assumption and mathematicians increasingly believed that maths is just a system for proving theorems from a set of axioms and inference rules, irrespective of whether the axioms are objectively true or the inference rules are objectively valid. This new abstract conception of maths prompted questions about its integrity – specifically, whether it’s ‘consistent’ and ‘complete’.
The Search for Consistency and Completeness
Consistency and Completeness
A logical system is consistent if it’s impossible to derive both a statement and its negation and it’s complete if every true statement can be derived. Sudoku is an example of a simple logical system and it can be used to demonstrate consistency and completeness.
Consistency
This is sudoku is inconsistent:
If we try to solve the cell in the top left-hand corner, X, using the 5 in the 3x3 region to the right of X we can deduce that X must be 5 (below left), whereas if we try to solve X using the 5 in the region below X we can deduce that X can’t be 5 (below right). In other words, the theorems X=5 and ~(X=5) can both be derived.
A sudoku is consistent if it’s impossible to deduce that a cell must contain a given number and also that it can’t contain that number.
Completeness
This sudoku is incomplete:
There is only one configuration of numbers this sudoku can have (below left) but it can’t be derived from the given numbers above. Therefore, X=9 is true but it’s not a theorem.
A sudoku is complete if every number can be derived from the numbers given at the outset.
Modelling
The first method that was used to establish consistency and completeness was to find a model that the axioms of the system could be compared with. For example, consider these axioms about two sets, X and Y.
1. Any two elements of X are contained in just one element of Y.
2. No element of X is contained in more than two elements of Y.
3. The elements of X are not all contained in a single element of Y.
4. Any two elements of X contain just one element of Y.
5. No element of X contains more than two elements of Y.
Using conventional inference rules, theorems can be derived from these axioms and it can be shown that the system is consistent and complete by modelling it geometrically – specifically, using a triangle where the elements of X are the vertices of a triangle and the elements of Y are the sides of the triangle.
As the diagram shows, the model applies all of the axioms without contradiction. This appears to demonstrate the consistency and completeness of the system but it actually shifts the question from sets X and Y to geometry. In other words, the axioms and inference rules of sets X and Y are only consistent and complete if the axioms and inference rules of geometry are consistent and complete. Using another model to prove the consistency and completeness of geometry would then shift the question to that model, and so on in an infinite regress.
Hilbert’s Program
After the failure of modelling, the German mathematician David Hilbert tried to prove consistency and completeness by ‘formalising’ maths. Formalisation makes a logical system unambiguous by creating an explicit vocabulary of elementary signs, providing clear definitions of the signs, establishing clear grammatical rules for combining the signs to create logical statements, proposing a set of axioms and establishing clear rules for deriving theorems from the axioms.
In a formalised system, the symbols, axioms, inference rules, etc. have no inherent meaning and are instead defined functionally in terms of how they can be used. This is similar to chess, in which the pieces have no real meaning but are governed by clear rules about how they can move.
Hilbert believed that formalisation could be used to prove consistency and completeness in part because it removed ambiguity from maths but also because it allowed statements about the system itself to be made rigorously. These statements are called ‘meta’ statements. For example, whereas ‘2+3=5’ is a mathematical statement, ‘2 is the first symbol in the formula “2+3=5”’ is a meta-mathematical statement. Hilbert believed meta statements could be used to prove consistency and completeness because the statements ‘this system is consistent’ and ‘this system is complete’ are meta statements which could be expressed (and hopefully proven) within the logical system. Formalising maths and deriving a meta-mathematical proof of its consistency and completeness became known as Hilbert’s Program.
An Example of a Successful Proof of Consistency
Principia Mathematica
In 1910, Bertrand Russell and Alfred North Whitehead published ‘Principia Mathematica’ in which they formalised arithmetic using a comprehensive symbolic notation which could be used to express every arithmetical statement. Elsewhere in Principia Mathematica, Russell and Whitehead formalised a simpler logical system called propositional logic and in 1921 Emil Post proved that the system was consistent. Here is the formalisation of propositional logic contained in Principia Mathematica:
Vocabulary
The first four elementary signs are called connectives because they connect the variables and formulas together.
Formation Rules
Inference Rules
Axioms
Proof of Consistency
The Principle of Explosion
This formula can be derived from the axioms:
Suppose the calculus is inconsistent and the formulas S and its negation ~S can both be derived. That would give us these three theorems:
Using the Rule of Substitution (the variables in any formula can be substituted for other variables and formulas, provided the same substitution is made for each occurrence of the variable) we can substitute S and ~S for p and ~p in the formula to give:
Then we can use the Rule of Detachment (from S1 and ‘if S1 then S2’ we can derive S2) to make two transformations:
The Rule of Substitution can then be used to substitute the theorem q for any formula. So, if an arbitrary formula, S, and its negation, ~S, are both theorems then we can derive every formula in the calculus. In other words, if the calculus is inconsistent then every formula is a theorem. This is known as the Principle of Explosion or ‘ex falso quodlibet’ – from a falsehood, anything follows. Therefore, proving the consistency of the calculus is equivalent to proving that at least one formula can’t be derived from the axioms.
This can be done by finding a property of the formulas which satisfies three criteria. First, it is possessed by all four axioms. Second, it is ‘hereditary’ under both inference rules – i.e., if a formula has the property then any theorem derived from it using the inference rules also has it. Third, it isn’t possessed by every formula – i.e., there is a grammatically valid formula which doesn’t have the property. In other words, if we can find a property that is common to all the axioms, common to all of the theorems which can be derived from the axioms using the inference rules but not common to all of the formulas which could possibly be created, we can prove that the system is consistent.
This seems complicated but it’s just a way to show that there are formulas which can’t be derived from the axioms using the inference rules. If there is a property which all of the axioms have and which they pass to all the theorems which can be derived from them using the inference rules and it is possible to create a formula which doesn’t have the property, then that formula can’t be derived from the axioms. Since every formula can be derived from the axioms of an inconsistent logical system, this is equivalent to proving the system is consistent.
A property that satisfies these three criteria is being ‘tautologous’. A tautology is a statement which is true regardless of whether its variables are true. For example, ‘it is either raining or it is not raining’ is obviously a tautology because it’s true regardless of whether it’s raining or not. In formal logic, this statement is written as pV~p – i.e., ‘either p is true or not-p is true’.
We can determine whether a formula is a tautology using truth tables. Truth tables show whether a formula is true or false for each of the different combinations of the truth statuses of the variables it contains. For example, here is the truth table for the formula p.q – i.e., ‘p is true and q is true’:
As we can see, the formula p.q is only true when the variables p and q are both true, so it isn’t a tautology. On the other hand, here is the truth table for the tautologous formula pV~p – i.e., ‘either p is true or not-p is true’:
Since a tautology is true regardless of the truth of the variables it contains, the final column of the truth table of a tautology will only contain the word ‘True’.
Post’s Proof
The Axioms are Tautologies
Here are the truth tables for the four axioms:
(For an explanation of why formulas of the form ‘if S1 then S2’ are true if S1 is false, see this footnote.)2
So, all four axioms are tautologies.
Being Tautologous is Hereditary Under the Inference Rules
The Rule of Substitution
If being tautologous is hereditary under the Rule of Substitution then if a formula is a tautology, any formula obtained through a substitution of its variables must also be a tautology. Suppose Sx is the tautology pV~p – i.e., ‘either p is true or p isn’t true’ – and we substitute p for qVr – i.e., ‘either q is true or r is true’ – which produces Sy – (qVr)V(~ (qVr)). Choose an arbitrary set of truth values for the variables of Sy – Ty. Then use those truth values to create a set of truth values for the variables of Sx – Tx. In other words, the truth value of qVr in Ty is used as the truth value of p in Tx – e.g., if qVr is true then so is p and if qVr is false then so is p. If Sx is true under Tx, then Sy must be true under Ty because the truth values of their variables are the same and their connectives are the same. This means that if Sx is a tautology then Sy must also be a tautology because Sx is true under all interpretations of Tx (by definition, since Sx is a tautology) so Sy must also be true under all interpretations of Ty. Therefore, being tautologous is hereditary under the Rule of Substitution.
The Rule of Detachment
If being tautologous is hereditary under the Rule of Detachment then if S1 and ‘if S1 then S2’ are both tautologies, S2 must also be a tautology. This can be proven by contradiction. Suppose S1 and ‘if S1 then S2’ are tautologies but S2 isn’t. That means that for at least one set of truth values of S2’s variables, T2, S2 isn’t true. S1 is a tautology, so it must be true under all sets of truth values of its variables, including T2. However, as the truth table below shows, if S1 is true and S2 is false then ‘if S1 then S2’ is false but that contradicts the fact that ‘if S1 then S2’ has been stated to be a tautology. Therefore, if the formulas S1 and ‘if S1 then S2’ are both tautologies then S2 must also be a tautology which means being tautologous is hereditary under the Rule of Detachment.
So, being tautologous is hereditary under both inference rules.
There are Formulas which aren’t Tautologies
As the truth table below shows, the formula pVq isn’t a tautology. When p and q are both false, pVq is also false.
So, the formula pVq isn’t a tautology. Therefore, it couldn’t have been derived from the axioms using the transformation rules. This means that not every formula can be derived and therefore the calculus is consistent.
Propositional Logic is Consistent
So, propositional logic can be proven to be consistent. Unfortunately, it is a very basic logical system which can’t be used support arithmetic. However, it gave mathematicians hope that more sophisticated logical systems could also be proven to be consistent. However, at the start of the twentieth century a French mathematician called Jules Richard created a paradox which later inspired Gödel to prove that more sophisticated logical systems can’t be proven to be either consistent or complete.
Richard’s Paradox
In 1905, Jules Richard published ‘Richard’s Paradox’. Richard proposed a system for cataloguing the various arithmetical properties that natural numbers can have. For example, ‘divisible by two’ (i.e., even), ‘divisible by exactly two natural numbers, one and itself’ (i.e., prime), ‘the product of two identical natural numbers’ (i.e., square), and so on.
These definitions can then be ordered alphabetically. Here is an example with only five definitions:
‘Divisible by exactly two natural numbers, one and itself’ – i.e., prime.
‘Divisible by two’ – i.e., even.
‘Not divisible by two’ – i.e., odd.
‘The product of two identical natural numbers’ – i.e., square.
‘The sum of its proper divisors’ – i.e., perfect.
In this list each definition corresponds to a unique natural number – i.e., its position alphabetically. Some of these numbers have the property described in their corresponding definitions and some don’t. For example, the numbers 1 and 5 don’t have the property described in their corresponding definition because 1 isn’t prime, and 5 isn’t perfect, whereas the numbers 2, 3 and 4 do have the property described in their corresponding definitions because 2 is even, 3 is odd and 4 is square.
Richard then defined a natural number which is not described by its corresponding definition as ‘Richardian’. In the list above, the natural numbers 1 and 5 are Richardian but 2, 3 and 4 aren’t. This makes being Richardian a property of natural numbers, so it can be included in the list:
‘Divisible by exactly two natural numbers, one and itself’ – i.e., prime.
‘Divisible by two’ – i.e., even.
‘Not described by its corresponding definition’ – i.e., Richardian.
‘Not divisible by two’ – i.e., odd.
‘The product of two identical natural numbers’ – i.e., square.
‘The sum of its proper divisors’ – i.e., perfect.
However, this leads to the paradoxical question of whether 3 is Richardian. Being Richardian means that the number isn’t described by its corresponding definition but 3’s corresponding definition is ‘not described by its corresponding definition’, so if 3 isn’t described by its corresponding definition then it is and if it is described by its corresponding definition then it isn’t. In other words, 3 is Richardian if, and only if, 3 isn’t Richardian. This appears to show that there are mathematical statements – such as ‘3 is Richardian’ – whose truth can’t be decided mathematically.
However, there is a flaw in Richard’s Paradox. The definitions are supposed to arithmetical properties but being Richardian is a meta-mathematical property. In other words, being Richardian isn’t a property of natural numbers in terms of other natural numbers, it’s a property of natural numbers in terms of a system for cataloguing their properties. So, Richard’s Paradox isn’t actually a paradox but its argument inspired Gödel and he set out to create a system which could make it logically rigorous.
Gödel’s Proofs
Gödel Numbering
To make Richard’s Paradox rigorous, Gödel developed system called ‘Gödel Numbering’ to map logic onto arithmetic. As we saw earlier, mapping translates one mathematical system into another. For example, geometry can be mapped onto algebra using coordinate systems and equations, allowing geometric shapes to be described algebraically.
Gödel used the formalised calculus created by Russell and Whitehead (which we’ll call ‘PM’ for Principia Mathematica) and assigned a number (called a Gödel Number) to each of the elementary signs in the vocabulary of PM:
Gödel also assigned a Gödel Number to three different types of variable – numerical variables (which can be substituted for numbers), sentential variables (which can be substituted for propositions) and predicate variables (which can be substituted for properties):
Gödel Numbering uses this system to convert any formula into a unique number by raising the first prime number to the power of the Gödel Number of the first symbol in the formula, raising the second prime number to the power of the Gödel Number of the second symbol in the formula, and so on, and then multiplying all the numbers together.
For example, the formula ‘0=0’ can be converted into a Gödel Number like this:
The Fundamental Theorem of Arithmetic proves that every number greater than 1 is the product of a unique collection of primes. For example, 12 is the product of 2, 2 and 3, so its prime factorisation is 22 times 3, and there is no other arrangement of primes that 12 can be factorised into. This means that every formula in PM corresponds to a unique Gödel Number and each Gödel Number corresponds to a unique formula in PM. For example, Gödel Numbering encodes the formula ‘0=0’ as 26 times 35 times 56 which equals the Gödel Number 243,000,000 and the Gödel Number 243,000,000 can be factorised to 26 times 35 times 56 and decoded to give the formula ‘0=0’. Because Gödel Numbering maps logic onto arithmetic, metalogical statements can be proved arithmetically. For example, the metalogical statement ‘0 is the first symbol in the formula 0=0’ can be proved arithmetically in terms of the Gödel Numbers of ‘0’ and ‘0=0’. In PM, ‘0=0’ has the Gödel number 243,000,000 and ‘0’ has the Gödel number 6, so if ‘0’ is the first symbol in the formula ‘0=0’ then the formula’s Gödel number must have 26 as factor, but no higher power of 2. Therefore, proving the meta-mathematical statement ‘0 is the first symbol in the formula 0=0’ is equivalent to proving the arithmetical statement ‘26 is a factor of 243,000,000 but 27 isn’t.’
The Demonstration and Substitution Formulas
Gödel then proposed two types of formula – the demonstration formula and the substitution formula:
Demonstration Formula
The demonstration formula states that a sequence of formulas proves another formula. For example, consider the demonstration formula, D:
D means ‘The sequence of formulas with Gödel Number x proves the formula with Gödel Number y.’ In other words, ‘y is a theorem in PM’.
Substitution Formula
The substitution formula inserts a formula’s Gödel Number into the formula itself. For example, consider the formula S:
S means ‘there is a number x such that x is the immediate successor of y.’ This formula has a Gödel Number, GN(S), and we can substitute y for GN(S) and rewrite S like this:
Which means ‘there is a number x such that x is the immediate successor of the Gödel Number of the formula S.’
The Incompleteness Theorems
The Gödel Sentence
Gödel used the demonstration and substitution formulas to create the formula, G:
In English, G means ‘There is no number x such that x is the Gödel Number of a sequence of formulas which proves the formula with Gödel Number GN(G).’ In other words, G means ‘G can’t be proven’.
The formula is known as the Gödel Sentence and Gödel used it to prove both Incompleteness Theorems.
Gödel’s First Incompleteness Theorem
Here’s a breakdown of G and its negation ~G:
This leads to the paradoxical question of whether G can be proven. As the ‘Plain English’ column of the table above shows, if G can be proven then ~G is true. So, G is true if, and only if, ~G is true. That means that if PM is consistent, then G is undecidable – i.e., G can’t be proven or disproven.
This is significant because G claims that it can’t be proven. In other words, the formula corresponding to the statement ‘This formula can’t be proven’ can’t be proven. So, G represents a true statement but it can’t be proven in PM if PM is consistent. Therefore, not every true formula is a theorem in PM if PM is consistent. So, if PM is consistent then PM is incomplete.
Gödel’s Second Incompleteness Theorem
Gödel then took the statement ‘If PM is consistent then G can’t be proven’ and converted it into a formula of PM.
As we saw during the proof of the consistency of propositional logic, if a logical system is inconsistent then every formula is a theorem, so proving that PM is consistent is equivalent to proving that there is at least one formula in PM which isn’t a theorem. So, we can convert the statement ‘PM is consistent’ into the formula C:
In English, C means ‘There is a formula with the Gödel Number y such that there is no sequence of formulas with the Gödel Number x which proves y’.
The formula corresponding to the statement ‘G can’t be proven’ is the formula G itself. So, the statement ‘If PM is consistent then G can’t be proven’ can be written as:
The first Incompleteness Theorem proved that the statement ‘If PM is consistent then G can’t be proven’ can be shown outside of PM but Gödel showed that ‘if C then G’ is also a theorem in PM. Gödel then used the theorem ‘if C then G’ to show that a proof of the consistency of PM can’t be derived in PM:
C corresponds to the statement ‘PM is consistent’, so proving in PM that PM is consistent means proving that C is a theorem of PM. If C is a theorem in PM, we would have the theorems C and ‘if C then G’. From C and ‘if C then G’ we can use the Rule of Detachment (from S1 and ‘if S1 then S2’, we can derive S2) to derive G. However, as we saw during the proof of the first Incompleteness Theorem, if G is true then ~G is also true, which means that PM isn’t consistent. Therefore, C can’t be a theorem in PM if PM is consistent. In other words, the consistency of PM can’t be proven in PM.
Summary
Gödel’s Incompleteness Theorems showed that Hilbert’s Program is impossible. The first Incompleteness Theorem showed that if maths is consistent then there are true mathematical statements which can’t be proven mathematically and the second Incompleteness Theorem showed that the consistency of maths can’t be proven mathematically. Consequently, it’s impossible to formalise maths in a calculus which is both consistent and complete.
Examples of Incompleteness
The Gödel sentence feels quite contrived and if this was the only true but unprovable statement in formalised maths then the first Incompleteness Theorem would be relatively inconsequential. However, there have been several significant mathematical statements which have been discovered to be true but unprovable.
The Paris-Harrington Theorem in Peano Arithmetic
Peano Arithmetic is a formalisation of arithmetic developed by Giuseppe Peano in the late nineteenth century and it is one of the most commonly used formalisations of arithmetic.
The Paris-Harrington Theorem is an extension of Ramsey’s Theorem in combinatorics. Ramsey’s Theorem states that when connecting points on a graph with coloured lines, if there are enough points then there will always be a group of points connected to each other with the lines of the same colour. For example, when connecting six points with two colours, there will always be three points connected by lines of the same colour:
The Paris-Harrington Theorem is similar to Ramsey’s Theorem but instead of a given number of points connected by lines of the same colour, it states that if the points are numbered then there will always be a group of points connected by lines of the same colour for any group containing more points than the lowest numbered point in the group.
The Paris-Harrington Theorem is true but unproveable in Peano Arithmetic and it’s considered the first naturally occurring example of incompleteness in formalised maths.
Goodstein’s Theorem in Peano Arithmetic
Goodstein’s Theorem states that every ‘Goodstein sequence’ converges to 0. A Goodstein sequence is generated using the following rules:
1. Write the input in base 2.
2. Keep the exponents the same but increase the base by 1, then subtract 1.
3. Write the result in the new base.
4. Repeat steps 2 and 3.
For example, starting with the input 3 we get the following Goodstein sequence:
Goodstein’s Theorem is true but can’t be proven in Peano Arithmetic, although it can be proven in more powerful formalisations like Zermelo-Fraenkel Set Theory and Principia Mathematica.
The Continuum Hypothesis in Zermelo-Fraenkel Set Theory
In the early twentieth century, Ernst Zermelo and Abraham Fraenkel developed a formalisation of set theory known as Zermelo-Fraenkel Set Theory (ZFC). ZFC is one of the most commonly used formalisations of mathematics.
The Continuum Hypothesis conjectures that there is no infinite set whose cardinality (size, essentially) is between the set of natural numbers and the set of real numbers. In 1940, Gödel himself proved that the Continuum Hypothesis can’t be disproven in ZFC and in 1963 Paul Cohen proved that it can’t be proven in ZFC. So, either the Continuum Hypothesis is true but can’t be proven or there is an infinite set with a cardinality between the set of natural numbers and the set of real numbers whose existence can’t be proven.
Note
The majority of this is taken from Gödel’s Proof written by Newman and Nagel and revised by Douglas Hofstadter.
Gödel’s character makes a brief appearance in Oppenheimer during the scene where Oppenheimer first visits Einstein at Princeton. When Oppenheimer finds Einstein, he’s walking in the woods with Gödel and after Einstein leaves to talk to Oppenheimer he tells him ‘You know, some days Kurt refuses to eat. Even in Princeton he’s convinced the Nazis can poison his food.’
The statement ‘if S1 is true, then S2 is true’ is obviously true if both S1 and S2 and true and obviously false if S1 is true but S2 is false. However, it’s less obvious that it’s true if S1 is false and S2 is true or if both S1 and S2 are false. These are known as ‘vacuous truths’ which essentially means that they’re true because they can’t be proven false. The statement ‘if S1 is true, then S2 is true’ can’t be proven false because it begins ‘if S1 is true…’ but S1 isn’t true. Vacuous truths seem arbitrary but they are included in classical logic primarily because excluding them leads to logical contradictions.